where the constants \(a_1,a_2>0\)  depend on the properties of \(f\)  and \(X\) , but not directly on the dimension \(d\) . There are three specific settings in which the concentration bound (REF ) has been obtained: (i) \(X\)  is a symmetric Markov process and the function \(f\)  is bounded, (ii) \(X\)  satisfies assumptions as in
() and \(f\)  is Lipschitz, (iii) \(X\)  is a symmetric Markov process that satisfies a (local) Poincaré inequality and \(f\)  fulfils certain growth and integrability conditions. We remark that the handling of these three settings is quite different in nature and the user has to decide which setting is more convenient in a particular situation. In this paper we will deal with the second setting (although one could also try to derive similar results in the other two settings; we note however that even in the Ornstein-Uhlenbeck model the function \(f\)  is not bounded, so the setting (i) does not apply). In the Lipschitz setting concentration inequalities are strongly related to transportation cost-information inequalities via the famous \(T_p(C)\)  conditions; see e.g. [1]}, [2]}, [3]}, [4]} for more details.
